## Abstract

We expose some basic elements of a style of programming supported by functional languages like Haskell by relating them to a coherent set of notions and techniques from Curry’s work in combinatory logic and formal systems, and their algebraic and categorical interpretations. Our account takes the form of a commentary to a simple fragment of Haskell code attempting to isolate the conceptual sources of the linguistic abstractions involved.

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## Notes

- 1.
Also the MIT course notes on programming linguistics (Wozencraft and Evans 1971), strongly influenced by lectures held at MIT by Landin and Strachey, use some of Curry’s terminology, speaking of “obs” and their “representations”.

- 2.
Curry (1952, 1958) observed the analogy between formal systems and abstract algebras, pointing out also their main differences, namely the fact that in an algebra, “the elements are conceived as existing beforehand” , where in a formal system “what is given beforehand is not a set of elements but the atoms and operations, and the obs are generated from them” (Curry and Feys 1958, §1B1).

- 3.
As an aside, we point out that Giovanni Vailati, a collaborator of Peano, had already studied the language of algebra and its grammar, in “La grammatica dell’algebra” (Rivista di Psicologia Applicata,

**4**, 1908), to which Peano replied more than twenty years later with his “Algebra de Grammatica”, Schola et Vita, vol. V (1930) pp. 323–336, where he outlines an algebraic approach to grammar based on the categories of verb, noun, and adjective that is strongly reminiscent of the more successful subsequent attempts by Ajdukiewicz, Bar Hillel, and especially Lambek. - 4.
Some proviso is needed, however, on the correspondence between programming language constructs and logical and algebraic notions. For example, in languages with lazy pattern matching, like Haskell, the elements of type Nat are not in bijective correspondence with the natural numbers: in Haskell, we can define infty = Succ infty for which the compiler infers type Nat, which does not correspond to any natural number but can nevertheless be used significantly as an argument of functions without causing non-terminating behavior (see Bird (1998) for examples). Other expressions for which the type Nat can be inferred but which do not correspond to any natural number are introduced by defining bottom = bottom and then taking bottom, Succ bottom, Succ (Succ bottom), … The type Nat is more accurately modeled by a partially ordered set enjoying a special completeness property in the order-theoretic sense – a

*cpo*; here, in addition to natural numbers, there is an infinite totally ordered subset whose elements corresponding to elements of Nat that involve bottom, whose least upper bound is the element corresponding to infty. This structure is still an initial algebra, but in a suitable category of cpo’s (Freyd 1991). - 5.
Observe that also the terms “closure” and “formation” are ambiguous and may refer both to processes and to their results, exactly like “construction”.

- 6.
The interest of

*T*-algebras in a computational setting can also be seen from their use in the categorical investigations on classes of automata by Arbib, Manes, and several others, in the early 1970s. There, a central notion is that of*dynamics*that generalizes the transition function of an automaton*δ*:*X*×*Q*→*Q*, where*X*is the input alphabet and*Q*the set of states of the automaton. The observation that the construction*X*×⋅ is an endofunctor over the category of sets makes this notion of dynamics a special case of the general categorical definition of an algebra of an endofunctor \(T: {\mathscr{C}} \longrightarrow {\mathscr{C}}\). In the context of the categorical reconstruction of automata theory,*T*-algebras were usually studied through the*free*monad over*T*(see Arbib and Manes (1974) for an early survey of this field). Monads have come to play an important role in structuring Haskell programs, although through a different path, following pioneering work by Moggi, Spivey, and Wadler in the late 1980s. - 7.
- 8.
See Thompson (1991) for more on the application of the inversion principle to programming.

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## Acknowledgments

I am grateful to the anonymous referees for insightful comments that have led to a definite improvement of the original version. My warmest thanks also to Simone Martini for presenting the results of this paper at a project meeting that I could not attend.

## Funding

The preparation of this paper has been financially supported by project PROGRAMme ANR-17-CE38-0003-01 (principal investigator Liesbeth De Mol).

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Cardone, F. From Curry to Haskell.
*Philos. Technol.* (2020). https://doi.org/10.1007/s13347-019-00385-4

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### Keywords

- History of functional programming
- Formal systems
- Initial algebras
- Inversion principle